CN112560949A - Hyperspectral classification method based on multilevel statistical feature extraction - Google Patents

Hyperspectral classification method based on multilevel statistical feature extraction Download PDF

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CN112560949A
CN112560949A CN202011477380.XA CN202011477380A CN112560949A CN 112560949 A CN112560949 A CN 112560949A CN 202011477380 A CN202011477380 A CN 202011477380A CN 112560949 A CN112560949 A CN 112560949A
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CN112560949B (en
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李丹
吴汉杰
孔繁锵
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses a hyperspectral classification method based on multilevel statistical feature extraction. Then, multi-level statistical characteristics including average characteristics, covariance description characteristics and Gaussian characteristics are provided, spatial geometrical information, correlation information among different spectrums and change characteristics among space-spectrum information are fully extracted, and classification accuracy can be improved to a great extent. And moreover, a multi-level kernel function is designed, a multi-task kernel sparse representation classification model is constructed, three statistical characteristics are effectively fused, a classification task is completed, and classification precision and stability under a weak supervision condition are further improved. Compared with the latest similar hyperspectral classification method, the feature extraction classification method provided by the invention has higher classification precision and can be applied to the actual engineering fields of hyperspectral classification, space remote sensing, geological exploration, agricultural information monitoring, ocean and atmosphere monitoring, military reconnaissance and the like.

Description

Hyperspectral classification method based on multilevel statistical feature extraction
Technical Field
The invention relates to a hyperspectral classification method based on multilevel statistical feature extraction, and belongs to the technical field of hyperspectral image processing and application.
Background
The hyperspectral image not only contains abundant spatial information, but also contains spectral information of hundreds or even thousands of wave bands, and is successfully applied to various engineering fields such as military target identification, geological exploration, material detection, ocean atmosphere monitoring and the like. The hyperspectral classification is the most essential problem, and the aim of the hyperspectral classification is to classify each spectral pixel point into a specific category. In order to complete the classification task, classification methods such as SVM, MLR and the like are provided. However, these methods have two major problems: (1) the classification precision is lower under the weak supervision condition; (2) noise caused by spectral aliasing. To solve these two main problems, many feature extraction methods are proposed, which are dedicated to extracting spatial features in different aspects and improving classification accuracy. One of the methods is a manual feature extraction method, and includes an EMP feature extraction method, an LBP feature extraction method, a multi-feature extraction method and the like. Although these manual feature extraction methods improve the classification accuracy to a great extent, the classification accuracy is drastically reduced under the condition of few training samples (weak supervision). The other is a classification method based on a Convolutional Neural Network (CNN), and the space-spectrum features from simple to complex are automatically extracted through a series of deep network layers. Although these methods also improve the high spectral classification accuracy to a great extent, there are two disadvantages: (1) the deep neural network training needs a large amount of training samples, and under the weak supervision condition, the training samples are fewer, so that the classification precision of the CNN-based classification method is essentially limited; (2) the deep network has a large number of parameters to be trained, which causes great computational complexity and limits the practical application of the deep network. Therefore, how to extract the spatial-spectral features with better recognition ability and effectively improve the classification accuracy under the condition of weak supervision remains a challenging task.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: how to accurately extract space-spectrum characteristics with better identification power, a classification method which effectively integrates multi-level statistical characteristics is constructed, and the classification precision and stability are greatly improved under the condition of weak supervision.
The invention adopts the following technical scheme for solving the technical problems:
a hyperspectral classification method based on multilevel statistical feature extraction is used for identifying a superpixel neighborhood set for each pixel, and is beneficial to accurately extracting spatial context information. And moreover, multi-level statistical characteristics such as average characteristics, covariance description characteristics and Gaussian characteristics are extracted, and the classification accuracy and stability under the weak supervision condition are effectively improved. And a multi-task core classifier is constructed to effectively integrate multi-level statistical characteristics, so that the classification precision is further improved.
The method comprises the following steps:
step 1, performing minimum noise separation and conversion MNF operation on an original hyperspectral image X to obtain dimension reduction hyperspectral data F;
step 2, sequentially using Principal Component Analysis (PCA) and an entropy rate-based super-pixel over-segmentation method to generate a 2-D super-pixel map of X;
step 3, for each dimensionality reduction pixel in the F, finding M' pixels with minimum cosine similarity among corresponding superpixels of the dimensionality reduction pixels to form a superpixel neighborhood set of the dimensionality reduction pixels;
step 4, extracting average characteristics from the super pixel neighborhood set of each dimensionality reduction pixel in the F; average characteristic of ith dimensionality reduction pixel in F
Figure BDA0002836047660000021
Wherein f isi pFor the p-th pixel in the super-pixel neighborhood set of the dimensionality reduction pixel, i is 1,2, …, n is the total number of pixels in F;
step 5, extracting covariance description characteristics from a super-pixel neighborhood set of each dimensionality reduction pixel in the F; covariance characterization of ith dimension-reduced pixel in F
Figure BDA0002836047660000022
Step 6, extracting Gaussian features from the super-pixel neighborhood set of each dimensionality reduction pixel in the F; gaussian feature of ith dimension-reduced pixel in F
Figure BDA0002836047660000023
Wherein Q isiPassing formula Ci＝QiQi ICalculating to obtain;
step 7, based onMean-feature, vectorized covariance description feature vec (C)i) And vectorized Gaussian feature vec (Y)i) Obtaining a set of spatio-spectral features of X
Figure BDA0002836047660000024
Wherein
Figure BDA0002836047660000025
For the spatio-spectral feature of the ith pixel in X,
Figure BDA0002836047660000026
step 8, designing a kernel function respectively to be ZrThe average feature, the covariance description feature and the Gaussian feature in the space are mapped to a uniform high-dimensional Hilbert space to obtain a mapped space-spectrum feature set Z'rWherein the average feature corresponds to a kernel function of k1(·,*)＝(·)T(x), covariance describes the kernel function of the feature as k2(log (·), log (·)), and the kernel function corresponding to the gaussian feature is k3(·,*)＝trace(log(·),log(*))；
Step 9, from Z'rRandomly selecting N characteristics of C classes to construct a statistical characteristic space dictionary
Figure BDA0002836047660000027
Wherein
Figure BDA0002836047660000028
A sub-dictionary corresponding to the class C, wherein C is 1,2, …, and C is the number of classes;
step 10, establishing a multi-task kernel sparse representation classification model based on multi-level statistical characteristics
Figure BDA00028360476600000212
s.t.‖αr‖0K or less, wherein alphar*For optimal sparse vector, αrIs a sparse vector, K is sparsity, z'rIs Z'rA certain pixel characteristic of;
step 11,Solving the model in the step 10 by using a kernel orthogonal matching pursuit algorithm to obtain alphar*；
Step 12, calculating the reconstruction error, wherein the reconstruction error of the c-th class
Figure BDA0002836047660000029
Figure BDA00028360476600000210
Is composed of
Figure BDA00028360476600000211
A corresponding optimal sparse subvector;
step 13, according to formula label (z'r)＝arg minc＝1,2,…,Cwc(z′r) To give z'rA category of (1);
step 14, repeating the steps 10 to 13 to obtain Z'rThe class of each feature is obtained, namely the feature of each pixel in X is obtained, and a class label graph of X is output.
Further, the formula for calculating the cosine similarity in step 3 is as follows:
Figure BDA0002836047660000031
in the formula (f)i 1For the ith reduced pixel in F, Fi qIs fi 1The q-th pixel of the corresponding super pixel.
Further, the solving method in step 11 includes the following steps:
11-1) calculating k separatelyr(z′r,z′r),Γ＝kr(Dr,Dr) And K ═ Kr(z′r,Dr)；
11-2) initializing sparse vectors
Figure BDA0002836047660000032
Index collection
Figure BDA0002836047660000033
And the residual error | (ss)0‖2＝kr(Z′r,Z′r) Minimum margin of error ε;
11-3) when t is less than or equal to K or
Figure BDA0002836047660000034
The following cycle was performed:
11-3-1) calculating kr(z′r,Dr)-αTΓ(:,Δt) And selecting the position of the maximum value in the calculation result and recording the position as idtWherein, istIs an index set at the t iteration;
11-3-2)Δt＝[Δt,idt]；
11-3-3) calculating gammat＝Γ(Δt,Δt) And Kt＝K(Δt)；
11-3-4) updating sparse representation
Figure BDA0002836047660000035
11-3-5) updating residual
Figure BDA0002836047660000036
11-4) obtaining sparse vectors
Figure BDA0002836047660000037
Compared with the prior art, the invention adopting the technical scheme has the following technical effects:
(1) according to the method, a super-pixel neighborhood set is identified for each pixel, so that the spatial context information of the hyperspectral image can be utilized more accurately;
(2) the method extracts multi-level statistical characteristics, fully extracts space set information, correlation information among different spectrums and related change information of space-spectrum information from different aspects, and improves classification accuracy and stability under weak supervision conditions to a great extent;
(3) according to the invention, a multi-core function is designed and a multi-task core sparse representation classifier is embedded, so that three statistical characteristics are effectively fused, and the classification precision is further improved.
(4) Compared with other methods of the same type, the method provided by the invention can improve the classification precision to a great extent under the condition of weak supervision.
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FIG. 1 is a schematic flow diagram of the process of the present invention;
FIG. 2 is an experimental hyperspectral image, wherein (a) is a real hyperspectral image and (b) is a correct label map;
FIG. 3 is a graph of the overall classification accuracy OA for a method MSFE of the present invention when the number of superpixels increases from 20 to 120 and the number of superpixel neighborhood pixels increases from 100 to 300;
FIGS. 4 (a) to (h) are visual comparison graphs of the classification results of the experimental hyperspectral images by eight classification methods of the method of the invention, namely MSFE and SSHGDA, SFMKSRC, LCMR, SRST, ISSR-DCNN, MCMs-2DCNN and S-DMM, respectively;
FIG. 5 is a graph of the overall classification accuracy OA of the eight classification methods of MSFE and SSHGDA, SFMKSRC, LCMR, SRST, ISSR-DCNN, MCMs-2DCNN, and S-DMM according to the method of the present invention when the number of selected training samples in each class is increased from 3 to 10.
Detailed Description
The invention provides a multilevel statistical feature extraction method and application thereof in hyperspectral images. Firstly, for each dimension-reduced pixel point, a super-pixel neighborhood set most similar to the dimension-reduced pixel point is identified from the corresponding super-pixel, so that not only can the spatial context information be more accurately extracted, but also the calculation complexity can be reduced to a great extent. Then, multi-level statistical features including mean features, covariance description features, and gaussian features are extracted from the super-pixel neighborhood set of each pixel to represent spatio-spectral features of the original pixel. The three statistical characteristics can fully extract space geometric information, correlation information among different spectrums and variation characteristics among space-spectrum information from different aspects and are mutually complementary. Another advantage is that although the super-pixel neighborhood sets of different pixels are different in size, the multi-level statistical features of different pixels have the same size and can be uniformly used for classification. And then, constructing a multi-level kernel function, mapping the multi-level statistical features to a unified Hilbert space, embedding the multi-level kernel function to construct a multi-task kernel sparse representation classifier, and finishing classification tasks. The multi-task kernel sparse representation classifier provides a natural method for integrating multi-level statistical features, and classification accuracy and stability under a weak supervision condition are greatly improved.
Fig. 1 shows a flow chart of the present invention, and the specific flow is described as follows:
step 1, MNF operation is carried out on the original hyperspectral image X to obtain dimension reduction hyperspectral data F.
Step 2, generating a 2-D super-pixel map of the original hyperspectral image by using a PCA (principal component analysis) and entropy rate super-pixel over-segmentation method;
step 3, reducing the dimension of the pixel F in the Fi 1Using cosine similarity formula
Figure BDA0002836047660000041
Identifying M' pixels which are most similar to the super pixels (with the smallest cosine similarity) in the corresponding super pixels to form a super pixel neighborhood set
Figure BDA0002836047660000042
And when M 'is greater than the number M of all pixel points in the corresponding super pixel, M' is equal to M. i is 1,2, …, n is the total number of pixels in f (x).
Step 4, reducing the dimension of the pixel F in the Fi 1From its super-pixel neighborhood set
Figure BDA0002836047660000043
Extracting average characteristics
Figure BDA0002836047660000044
Step 5, reducing the dimension of the pixel F in the Fi 1From its super-pixel neighborhood set
Figure BDA0002836047660000051
Mid-extraction covariance characterization feature
Figure BDA0002836047660000052
Step 6, for each reduced dimension pixel fi 1From its super-pixel neighborhood set
Figure BDA0002836047660000053
Extracting Gauss characteristics
Figure BDA0002836047660000054
Wherein Q isiPassing formula Ci＝QiQi TAnd (6) calculating.
Step 7, describing the feature vec (C) based on the mean feature and vectorized covariancei) And vectorized Gaussian feature vec (Y)i) Obtaining a set of spatio-spectral features of X
Figure BDA0002836047660000055
Wherein
Figure BDA0002836047660000056
For the spatio-spectral feature of the ith pixel in X,
Figure BDA0002836047660000057
step 8, designing a kernel function respectively to be ZrThe average feature, the covariance description feature and the Gaussian feature in the space are mapped to a uniform high-dimensional Hilbert space to obtain a mapped space-spectrum feature set Z'rWherein the average feature corresponds to a kernel function of k1(·,*)＝(·)T(x), covariance describes the kernel function of the feature as k2(log (·), log (·)), and the kernel function corresponding to the gaussian feature is k3(·,*)＝trace(log(·),log(*))。
Step 9, from Z'rRandomly selecting N characteristics of C classes to construct a statistical characteristic space dictionary
Figure BDA0002836047660000058
Wherein
Figure BDA0002836047660000059
The sub-dictionary corresponding to the class C is given by C1, 2, …, and C is the number of classes.
Step 10, establishing a multi-task kernel sparse representation classification model based on multi-level statistical characteristics
Figure BDA00028360476600000516
s.t.‖αr‖0K is less than or equal to K. Wherein K is sparsity.
Step 11, solving the model by using a kernel orthogonal matching pursuit algorithm to obtain alphar*。
Step 12, calculating the reconstruction error, wherein the reconstruction error of the c-th class
Figure BDA00028360476600000510
Figure BDA00028360476600000511
Is composed of
Figure BDA00028360476600000512
The corresponding optimal sparse subvector.
Step 13, according to formula label (z'r)＝arg minc＝1,2,…,Cwc(z′r) To give z'rThe category (2).
Step 14, repeating the steps 10 to 13 circularly to obtain Z'rThe class of each feature is obtained, namely the feature of each pixel in X is obtained, and a class label graph of X is output.
The above-mentioned kernel orthogonal matching pursuit algorithm in step 11 solves the sparse representation αr*The algorithm flow of (1) is as follows:
11-1) calculating k separatelyr(z′r,z′r),Γ＝kr(Dr,Dr) And K ═ Kr(z′r,Dr)；
11-2) initializing thinSparse vector
Figure BDA00028360476600000513
Index collection
Figure BDA00028360476600000514
And the residual error | (ss)0‖2＝kr(Z′r,Z′r) Minimum margin of error ε;
11-3) when t is less than or equal to K or
Figure BDA00028360476600000515
The following cycle was performed:
11-3-1) calculating kr(z′r,Dr)-αTΓ(:,Δt) And selecting the position of the maximum value in the calculation result and recording the position as idtWherein, istIs an index set at the t iteration;
11-3-2)Δt＝[Δt,idt]；
11-3-3) calculating gammat＝Γ(Δt,Δt) And Kt＝K(Δt)；
11-3-4) updating sparse representation
Figure BDA0002836047660000061
11-3-5) updating residual
Figure BDA0002836047660000062
11-4) obtaining sparse vectors
Figure BDA0002836047660000063
In order to better embody the advantages of the multi-level statistical feature extraction and the application thereof in hyperspectral classification, a specific example is combined below to compare the method disclosed by the invention with the newly proposed feature extraction and classification methods SSHGDA, SFMKSRC, LCMR, SRST, ISSR-DCNN, MCMs-2DCNN and S-DMM.
The comparison method is as follows: classifying the real hyperspectral image Salinas (corresponding to the correct label image in FIG. 2, the image (b) in FIG. 2) shown in FIG. 2, randomly selecting 5 pixel points in each type of pixel set as training samples, and using the rest pixel points as test sample sets. Firstly, when the number of the super pixels is increased from 20 to 120 and the number of the super pixel neighborhood pixels is increased from 100 to 300, the MSFE is classified by using the method of the invention to obtain the optimal parameter setting of the MSFE. And under the condition of the number of the training samples, the classification results achieved by the 8 characteristic extraction and classification methods are compared. The classification result is expressed using the overall classification accuracy (OA), the average classification accuracy (AA), the k coefficient, and the classification accuracy of each class. In order to further analyze the influence of different training sample quantities on the method, when the training sample quantity is increased from 3 pixels randomly selected from each class to 10 pixels selected from each class, 8-classification methods are respectively used for carrying out classification experiments on the experimental images, and OA evaluation is used for classification results.
FIG. 3 is a graph of the overall classification accuracy OA for a method MSFE of the present invention when the number of superpixels increases from 20 to 120 and the number of superpixel neighborhood pixels increases from 100 to 300. As can be seen from fig. 3, the classification accuracy of MSFE increases as the number of superpixels increases from 20 to 60. With the continuous increase of the number of super pixels, the classification accuracy of the MSFE is reduced to a certain extent. This is mainly because the larger number of superpixels makes the spatial information not fully utilized, while the smaller number of superpixels introduces different classes of pixel points for each superpixel. It can also be seen from fig. 3 that as the number of super-pixel neighborhood pixels grows from 100 to 220, the classification accuracy of MSFE increases. With the continuous increase of the number of the super-pixel neighborhood pixels, the classification accuracy of the MSFE is reduced. The main reason is that the larger number of super-pixel neighborhood pixels introduces different types of pixels, which causes classification errors, while the smaller number of super-pixel neighborhood pixels cannot fully utilize spatial information. Therefore, the method of the invention has the following optimum parameters: l60 and M220.
Table 1 shows comparison simulation results of OA, AA and kappa coefficients and various classification accuracies of the hyperspectral images of the experiment by 8 feature extraction and classification methods. As can be seen from Table 1, the MSFE of the method of the present invention obtained the highest OA, AA and kappa numbers, i.e., the best classification results, and was at least 5% higher than the other 7 newly proposed classification methods. FIGS. 4 (a) to (h) are visual comparison graphs of the classification results of the experimental hyperspectral images by eight classification methods of the method of the invention, namely MSFE and SSHGDA, SFMKSRC, LCMR, SRST, ISSR-DCNN, MCMs-2DCNN and S-DMM, respectively. As can be seen from fig. 4, the MSFE of the method of the present invention can provide the best classification result.
TABLE 1 comparison simulation results of OA, AA, kappa coefficients and each type of precision of experimental images by 8 methods when the number of each type of training samples is 5
Figure BDA0002836047660000071
FIG. 5 is a graph of the overall classification accuracy OA of the eight classification methods of MSFE and SSHGDA, SFMKSRC, LCMR, SRST, ISSR-DCNN, MCMs-2DCNN, and S-DMM according to the method of the present invention when the number of selected training samples in each class is increased from 3 to 10. As can be seen from fig. 5, the MSFE achieves the highest classification accuracy when the number of selected training samples in each class is increased from 3 to 10. And under the condition of extremely small number of training samples (weak supervision), the advantages of the MSFE are more obvious than those of other contrast classification methods.
In summary, the present invention identifies the super-pixel neighborhood set for each pixel, and accurately utilizes the spatial context information, which is beneficial to improve the classification precision. Then, multi-level statistical features including average features, covariance description features and Gaussian features are extracted, spatial geometrical information, correlation information among different spectral bands and related change information among space and spectrum are fully extracted, and classification accuracy is greatly improved. And moreover, a multi-level kernel function is designed, a multi-task kernel sparse representation classifier is constructed, multi-level statistical characteristics are effectively fused, classification tasks are completed, and classification precision is further improved.
The above examples are intended to illustrate the present invention, but not to limit the present invention. Any person skilled in the art will realize that changes or substitutions can be easily conceived and reduced to the above embodiments within the technical scope of the present disclosure, and that modifications to the above embodiments will fall within the scope of the claims of the present invention.

Claims (3)

1. The hyperspectral classification method based on multilevel statistical feature extraction is characterized by comprising the following steps of:
step 1, performing minimum noise separation and conversion MNF operation on an original hyperspectral image X to obtain dimension reduction hyperspectral data F;
step 2, sequentially using Principal Component Analysis (PCA) and an entropy rate-based super-pixel over-segmentation method to generate a 2-D super-pixel map of X;
step 3, for each dimensionality reduction pixel in the F, finding M' pixels with minimum cosine similarity among corresponding superpixels of the dimensionality reduction pixels to form a superpixel neighborhood set of the dimensionality reduction pixels;
step 4, extracting average characteristics from the super pixel neighborhood set of each dimensionality reduction pixel in the F; average characteristic of ith dimensionality reduction pixel in F
Figure FDA0002836047650000011
Wherein f isi pFor the p-th pixel in the super-pixel neighborhood set of the dimensionality reduction pixel, i is 1,2, …, n is the total number of pixels in F;
step 5, extracting covariance description characteristics from a super-pixel neighborhood set of each dimensionality reduction pixel in the F; covariance characterization of ith dimension-reduced pixel in F
Figure FDA0002836047650000012
Step 6, extracting Gaussian features from the super-pixel neighborhood set of each dimensionality reduction pixel in the F; gaussian feature of ith dimension-reduced pixel in F
Figure FDA0002836047650000013
Wherein Q isiPassing through type
Figure FDA00028360476500000113
Calculating to obtain;
step 7, describing the feature vec (C) based on the mean feature and vectorized covariancei) And vectorized Gaussian feature vec (Y)i) Obtaining a set of spatio-spectral features of X
Figure FDA0002836047650000014
Wherein
Figure FDA0002836047650000015
For the spatio-spectral feature of the ith pixel in X,
Figure FDA0002836047650000016
step 8, designing a kernel function respectively to be ZrThe average feature, the covariance description feature and the Gaussian feature in the space are mapped to a uniform high-dimensional Hilbert space to obtain a mapped space-spectrum feature set Z'rWherein the average feature corresponds to a kernel function of
Figure FDA00028360476500000114
The covariance describes the kernel function corresponding to the feature as k2(log (·), log (·)), and the kernel function corresponding to the gaussian feature is k3(·,*)＝trace(log(·),log(*))；
Step 9, from Z'rRandomly selecting N characteristics of C classes to construct a statistical characteristic space dictionary
Figure FDA0002836047650000017
Wherein
Figure FDA0002836047650000018
A sub-dictionary corresponding to the class C, wherein C is 1,2, …, and C is the number of classes;
step 10, establishing a multi-task kernel sparse representation classification model based on multi-level statistical characteristics
Figure FDA0002836047650000019
s.t.‖αr‖0K or less, wherein alphar*For optimal sparse vector, αrIs a sparse vector, K is sparsity, z'rIs Z'rA certain pixel characteristic of;
step 11, solving the model in the step 10 by using a kernel orthogonal matching pursuit algorithm to obtain alphar*；
Step 12, calculating the reconstruction error, wherein the reconstruction error of the c-th class
Figure FDA00028360476500000110
Figure FDA00028360476500000111
Is composed of
Figure FDA00028360476500000112
A corresponding optimal sparse subvector;
step 13, according to formula label (z'r)＝argminc＝1,2,…,Cwc(z′r) To give z'rA category of (1);
step 14, repeating the steps 10 to 13 to obtain Z'rThe class of each feature is obtained, namely the feature of each pixel in X is obtained, and a class label graph of X is output.
2. The hyperspectral classification method based on multistage statistical feature extraction according to claim 1, wherein the cosine similarity in step 3 is calculated by the formula:
Figure FDA0002836047650000021
in the formula (f)i 1For the ith reduced pixel in F, Fi qIs fi 1The q-th pixel of the corresponding super pixel.
3. The hyperspectral classification method based on multistage statistical feature extraction according to claim 1, wherein the solving method in step 11 comprises the following steps:
11-1) calculating k separatelyr(z′r,z′r),Γ＝kr(Dr,Dr) And K ═ Kr(z′r,Dr)；
11-2) initializing sparse vectors
Figure FDA0002836047650000022
Index collection
Figure FDA0002836047650000023
And the residual error | (ss)0‖2＝kr(Z′r,Z′r) Minimum margin of error ε;
11-3) when t is less than or equal to K or
Figure FDA0002836047650000024
The following cycle was performed:
11-3-1) calculation
Figure FDA0002836047650000028
And selecting the position of the maximum value in the calculation result and recording the position as idtWherein, istIs an index set at the t iteration;
11-3-2)Δt＝[Δt,idt]；
11-3-3) calculating gammat＝Γ(Δt,Δt) And Kt＝K(Δt)；
11-3-4) updating sparse representation
Figure FDA0002836047650000025
11-3-5) updating residual
Figure FDA0002836047650000026
11-4) obtaining sparse vectors
Figure FDA0002836047650000027
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